Combinatorial Proof

From the way you put the question, I assume that part (a) is no problem.
Any string in is a string in with one of 1,2, or 3 added. To avoid repetition we can use only two of three. Therefore , because the given result easily follows.

EDIT:
Clearly any string in with the last number removed is a string in . Therefore, taking any string in that is not in and adding the same number as it begins with gives a string in . Thus .
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